Integration

Integration by Parts

Find ∫x · e^x dx using integration by parts with u = x, dv = e^x dx, giving x · e^x - e^x + C.

integration-by-parts Integration by parts uses the formula ∫u dv = uv - ∫v du. Choose u and dv so that the remaining integral is simpler.

Example

Choose u and dv, then apply uv minus the integral of v du.

highlighted = computed this step

Step 1 — Set up

Set up the product integral.

∫xex dx\int xe^{x} \,dx

Step 2 — Choose parts

Choose u and dv.

u=xdv=ex dxu= \hlmath{x} \quad dv= \hlmath{e^{x}} \,dx

Step 3 — Compute du and v

Compute du and v.

du=dxv=exdu=dx \quad v= \hlmath{e^{x}}

Step 4 — Apply formula

Use uv minus the integral of v du.

∫u dv=uv−∫v du\int u\,dv=uv-\int v\,du

Step 5 — Result

State the result with plus C.

xex−ex+C\hlmath{xe^{x}-e^{x}} +C